""" ================================================================================ TOPOLOGICAL COOPER SCAFFOLD — SIMULATION v3.0 ================================================================================ Paper: "Topological Cooper Scaffold: Proximity-Induced Nodal-Line Protection as a Pathway Toward Elevated-Tc Superconductivity in MATBG" Author: Leon Sandler (leon@eliptum.com) Zenodo: https://doi.org/10.5281/zenodo.20821704 SCIENTIFIC QUESTION (v3 focus, per reviewer recommendation): How robust is the Topological Cooper Scaffold to realistic interface non-uniformity in the proximity exchange field? NEW IN v3 vs v2: - Interface uniformity parameter U_Δ ∈ [0,1] (1 = perfectly uniform) - Tc depends on VARIANCE of Δ_ex distribution, not only mean - Pair-breaking computed from upper tail of Δ_ex distribution - Scaffold benefit computed from mean Δ_ex - New figures: Fig10: Tc vs mean Δ_ex at multiple uniformity levels Fig11: Goldilocks window width vs interface uniformity U_Δ Fig12: Tc vs variance σ_Δ (robustness landscape) Fig13: Patterned nanoribbon geometry schematic Fig14: Fraction of k-space exceeding pair-breaking vs U_Δ PHYSICS OF U_Δ: A continuous cobalt film has spatially varying Δ_ex due to surface roughness, grain boundaries, and hBN thickness fluctuations. We model this as a Gaussian distribution of local exchange splittings: Δ_ex,local ~ N(Δ_mean, σ_Δ²) where σ_Δ = (1 - U_Δ) × Δ_mean (coefficient of variation scales with non-uniformity). Pair-breaking occurs wherever Δ_ex,local > Δ_pb (pair-breaking threshold). The fraction of k-space exceeding the threshold is: f_pb = P(Δ_ex > Δ_pb) = 1 - Φ((Δ_pb - Δ_mean)/σ_Δ) Effective Tc is reduced by this fraction: Tc_eff = Tc_uniform × (1 - α × f_pb) where α is the sensitivity of Tc to pair-broken k-space fraction. The scaffold benefit (S_spin suppression) acts on the MEAN Δ_ex, independent of the variance. Patterned nanoribbon geometry increases U_Δ by reducing spatial variance without changing mean Δ_ex. DISCLAIMER: All parameters are literature-derived estimates. U_Δ is a new parameter introduced to study robustness; its value for specific geometries requires experimental characterisation. Results are hypothesis-generating. ================================================================================ """ import numpy as np from scipy.optimize import brentq from scipy.integrate import quad from scipy.stats import norm import matplotlib matplotlib.use('Agg') import matplotlib.pyplot as plt from matplotlib.gridspec import GridSpec from matplotlib.patches import FancyArrow, Rectangle, FancyArrowPatch import matplotlib.patches as mpatches import warnings, os from datetime import datetime warnings.filterwarnings('ignore') OUT_DIR = os.path.dirname(os.path.abspath(__file__)) # ───────────────────────────────────────────────────────────────────────────── # SECTION 1 — PARAMETERS (unchanged from v2) # ───────────────────────────────────────────────────────────────────────────── K_B = 8.617e-5 # eV/K TC0 = 1.70 # K [Cao 2018] OMEGA_PH_MEV = 150.0 # meV [Ref 3] OMEGA_PH_K = OMEGA_PH_MEV*1e-3/K_B OMEGA_LOG_K = OMEGA_PH_K*0.85 MU_STAR = 0.10 DELTA_EV = 3.5*K_B*TC0/2 # eV (~Pauli threshold) F_IV = 0.60 E_SO_EV = 0.20e-3 J0_EV = 50e-3 HBN_NM = 0.33 XI_EX_NM = 0.51 DOS_FAC = 0.015 MU_B_EV = 5.788e-5 # ───────────────────────────────────────────────────────────────────────────── # SECTION 2 — NEW: INTERFACE UNIFORMITY MODEL # ───────────────────────────────────────────────────────────────────────────── # Uniformity parameter U_Δ ∈ [0,1] # U_Δ = 1: perfectly uniform Δ_ex (ideal) # U_Δ = 0: completely random Δ_ex (no spatial control) # Physical interpretation: # Continuous cobalt film: U_Δ ≈ 0.3–0.5 (rough estimate) # Patterned nanoribbon array (optimised): U_Δ ≈ 0.7–0.9 # Perfect single-crystal interface: U_Δ → 1 # Spatial variance of Δ_ex as fraction of mean: # σ_Δ = (1 - U_Δ) × Δ_mean # Coefficient of variation CV = σ_Δ/Δ_mean = (1 - U_Δ) def sigma_delta(dex_mean, U): """Spatial standard deviation of Δ_ex given mean and uniformity.""" return (1.0 - U) * dex_mean def frac_pair_broken(dex_mean, U, threshold=DELTA_EV): """ Fraction of k-space where local Δ_ex exceeds pair-breaking threshold. Δ_ex,local ~ N(dex_mean, σ_Δ²) f_pb = P(Δ_ex,local > threshold) """ if U >= 1.0: return 1.0 if dex_mean >= threshold else 0.0 sig = sigma_delta(dex_mean, U) if sig < 1e-12: return 1.0 if dex_mean >= threshold else 0.0 return 1.0 - norm.cdf(threshold, loc=dex_mean, scale=sig) # Sensitivity of Tc to fraction of pair-broken k-space # If f_pb fraction of k-space is locally above pair-breaking threshold, # those regions do not support pairing; effective Tc is reduced. # Linear model (conservative): Tc_eff = Tc_uniform × (1 - α × f_pb) ALPHA_PB = 1.5 # sensitivity: at f_pb=0.67 → Tc → 0 (reasonable for Cooper condensate) def tc_effective(dex_mean, U): """ Tc accounting for interface non-uniformity. Scaffold benefit acts on mean Δ_ex; pair-breaking acts on upper tail. """ if dex_mean <= 0: return TC0 # Step 1: Scaffold benefit from mean exchange splitting s = 1.0/np.sqrt(1.0+(dex_mean/E_SO_EV)**2) delta_mu = 0.20*F_IV*(1-s**2) mu_eff = max(MU_STAR - delta_mu, 0.01) # Step 2: AG pair-breaking from MEAN exchange field if dex_mean < DELTA_EV*0.10: ag = 1.0 - 0.05*(dex_mean/DELTA_EV) else: ag = max(0.0, 1.0-(np.pi/4)*(dex_mean/DELTA_EV)) # Step 3: Allen-Dynes Tc (uniform case) def ad(lam, mu=mu_eff): d = lam - mu*(1+0.62*lam) if d<=0: return 0 return (OMEGA_LOG_K/1.2)*np.exp(-1.04*(1+lam)/d) tc_unif = ad(LAM0)*ag # Step 4: Non-uniformity reduction from pair-broken k-space fraction f_pb = frac_pair_broken(dex_mean, U) tc_eff = tc_unif * max(0, 1.0 - ALPHA_PB*f_pb) return max(0.0, tc_eff) # Calibrate λ₀ def ad_base(lam): d = lam - MU_STAR*(1+0.62*lam) if d<=0: return 0 return (OMEGA_LOG_K/1.2)*np.exp(-1.04*(1+lam)/d) LAM0 = brentq(lambda l: ad_base(l)-TC0, 0.10, 0.60) def J_coupling(n): return J0_EV*np.exp(-n*HBN_NM/XI_EX_NM) def dex_n(n): return J_coupling(n)*DOS_FAC def sspin(dex): return 1.0/np.sqrt(1.0+(dex/E_SO_EV)**2) def hc2(dex): hc2_p = DELTA_EV/(np.sqrt(2)*MU_B_EV) if dex>=DELTA_EV: return hc2_p*0.05 s = sspin(dex) return hc2_p*(1.0+(1-s)*2.0) HC2_BASE = DELTA_EV/(np.sqrt(2)*MU_B_EV) # ───────────────────────────────────────────────────────────────────────────── # SECTION 3 — COMPUTE # ───────────────────────────────────────────────────────────────────────────── print("="*70) print("TOPOLOGICAL COOPER SCAFFOLD — SIMULATION v3.0") print(f"Scientific question: Robustness to interface non-uniformity") print(f"Run: {datetime.now().strftime('%Y-%m-%d %H:%M:%S')}") print("="*70) print(f"\nλ₀ = {LAM0:.4f} | Allen-Dynes check: {ad_base(LAM0):.4f}K ✓") DEX_ARR = np.linspace(0, DELTA_EV*5, 600) WEEKS = DEX_ARR*1000 # meV # Uniformity levels to compare U_LEVELS = [0.20, 0.40, 0.60, 0.80, 0.95, 1.00] U_LABELS = ['U=0.20\n(very rough)', 'U=0.40\n(rough)', 'U=0.60\n(moderate)', 'U=0.80\n(good)', 'U=0.95\n(nanoribbon)', 'U=1.00\n(perfect)'] U_COLORS = ['#c0392b', '#e67e22', '#f1c40f', '#2e8b57', '#1a6b8a', '#2E5FA3'] # Tc vs Δ_ex for each uniformity level TC_U = {u: np.array([tc_effective(d, u) for d in DEX_ARR]) for u in U_LEVELS} # Find peaks and window widths for each U results = {} for u in U_LEVELS: tc_arr = TC_U[u] win = tc_arr > TC0 if win.any(): opt_i = tc_arr.argmax() dex_o = DEX_ARR[opt_i] tc_o = tc_arr[opt_i] dtc_o = tc_o - TC0 w_lo = DEX_ARR[win][0] if win.any() else 0 w_hi = DEX_ARR[win][-1] if win.any() else 0 w_wid = (w_hi - w_lo)*1000 # meV else: dex_o, tc_o, dtc_o, w_lo, w_hi, w_wid = 0,TC0,0,0,0,0 results[u] = dict(dex_opt=dex_o, tc_opt=tc_o, dtc=dtc_o, win_lo=w_lo, win_hi=w_hi, win_width_mev=w_wid) print(f" U={u:.2f}: ΔTc_peak=+{dtc_o:.3f}K Goldilocks width={w_wid:.3f}meV") # Tc vs variance σ_Δ landscape (fixed mean = optimal from U=1) dex_opt_perfect = results[1.00]['dex_opt'] sigma_range = np.linspace(0, dex_opt_perfect*1.5, 200) # Convert σ to U: U = 1 - σ/dex_mean U_from_sigma = np.clip(1 - sigma_range/max(dex_opt_perfect,1e-9), 0, 1) tc_vs_sigma = np.array([tc_effective(dex_opt_perfect, u) for u in U_from_sigma]) # Fraction of k-space pair-broken vs U at optimal mean Δ_ex fpb_vs_U = np.array([frac_pair_broken(dex_opt_perfect, u) for u in np.linspace(0,1,200)]) # Spacer sweep at different uniformity levels SPACER_N = np.array([0,1,2,3,4,5]) TC_SP = {} for u in [0.40, 0.70, 0.95, 1.00]: TC_SP[u] = np.array([tc_effective(dex_n(n), u) for n in SPACER_N]) # Goldilocks window width vs U U_sweep = np.linspace(0.01, 1.00, 200) window_widths = [] for u in U_sweep: tc_arr = np.array([tc_effective(d, u) for d in DEX_ARR]) win = tc_arr > TC0 ww = (DEX_ARR[win][-1] - DEX_ARR[win][0])*1000 if win.sum()>1 else 0 window_widths.append(ww) window_widths = np.array(window_widths) # ───────────────────────────────────────────────────────────────────────────── # SECTION 4 — FIGURES (10-14 continuing from v2's Fig 1-9) # ───────────────────────────────────────────────────────────────────────────── C = {'bg':'#f7f9fb','grid':'#dce6ed','base':'#2E5FA3','warn':'#c0392b', 'good':'#2e8b57','mid':'#e67e22','pauli':'#c0392b','cob':'#8e44ad'} DIS = '⚠ Semi-microscopic model | U_Δ is a new parameter requiring experimental calibration | Not first-principles DFT' def sax(ax, t, xl, yl): ax.set_facecolor('white') ax.set_title(t, fontsize=10, fontweight='bold', color='#1F3864', pad=7) ax.set_xlabel(xl, fontsize=9); ax.set_ylabel(yl, fontsize=9) ax.grid(True, alpha=0.3, color=C['grid'], lw=0.7) for sp in ax.spines.values(): sp.set_color('#cccccc') ax.tick_params(labelsize=8.5) def wm(fig): fig.text(0.5,0.003,DIS,ha='center',fontsize=6.5,color='#c0392b',style='italic') # ── Fig 10: Tc vs Δ_ex at multiple uniformity levels ───────────────────────── fig10, ax = plt.subplots(figsize=(12,7)) fig10.patch.set_facecolor(C['bg']) ax.axhline(TC0, color='black', lw=1.8, ls='--', alpha=0.7, label=f'Bare Tc₀={TC0}K', zorder=3) ax.axvline(DELTA_EV*1000, color=C['warn'], lw=1.5, ls=':', alpha=0.8, label=f'~Pauli threshold ({DELTA_EV*1000:.2f}meV)') for u, col, lbl in zip(U_LEVELS, U_COLORS, U_LABELS): tc_arr = TC_U[u] ax.plot(DEX_ARR*1000, tc_arr, color=col, lw=2.2, label=f'{lbl} (ΔTc_peak=+{results[u]["dtc"]:.2f}K)') ax.fill_between(DEX_ARR*1000, TC0, TC_U[1.00], where=TC_U[1.00]>TC0, alpha=0.07, color=C['base'], zorder=0) sax(ax, 'Figure 10: Tc vs Mean Exchange Splitting at Multiple Interface Uniformity Levels\n' 'U_Δ=1.0: uniform (perfect); U_Δ=0.2: rough film. Key question: does the mechanism survive realistic imperfections?', 'Mean Δ_ex (meV)', 'Tc (K)') ax.set_xlim(0, DELTA_EV*4000); ax.set_ylim(-0.3, TC_U[1.00].max()*1.25+0.3) ax.legend(fontsize=8, loc='upper right', framealpha=0.9) ax.text(0.02,0.08, f'As U_Δ decreases from 1.0→0.2:\n' f'• Peak ΔTc decreases\n' f'• Goldilocks window NARROWS\n' f'• Effect vanishes at U_Δ≈0.2\n' f'Patterned nanoribbons target U_Δ≈0.7–0.9', transform=ax.transAxes, fontsize=8.5, bbox=dict(boxstyle='round',facecolor='white',alpha=0.9)) wm(fig10); plt.tight_layout(rect=[0,0.02,1,1]) fig10.savefig(os.path.join(OUT_DIR,'Fig10_Tc_vs_Uniformity.png'),dpi=180,bbox_inches='tight',facecolor=C['bg']) plt.close(); print(" ✓ Fig10") # ── Fig 11: Goldilocks window width vs U_Δ ─────────────────────────────────── fig11, axes = plt.subplots(1,2,figsize=(14,6)) fig11.patch.set_facecolor(C['bg']) ax = axes[0] ax.plot(U_sweep, window_widths, color=C['base'], lw=2.8, label='Goldilocks window width') ax.fill_between(U_sweep, 0, window_widths, alpha=0.15, color=C['base']) # Mark key uniformity levels for u, col, lbl in [(0.40,'#e67e22','Rough film'),(0.70,C['good'],'Moderate'),(0.95,'#1a6b8a','Nanoribbon est.')]: w = window_widths[np.argmin(np.abs(U_sweep-u))] ax.scatter([u],[w],color=col,s=80,zorder=6) ax.annotate(f'{lbl}\nU={u:.2f}\nw={w:.2f}meV', xy=(u,w),xytext=(u-0.12,w+0.02),fontsize=7.5,color=col, arrowprops=dict(arrowstyle='->',color=col)) ax.axvline(0.70, color=C['good'], lw=1.2, ls=':', alpha=0.7) ax.axvline(0.95, color='#1a6b8a', lw=1.2, ls=':', alpha=0.7) sax(ax,'Figure 11a: Goldilocks Window Width vs Interface Uniformity\nWider window = more experimental tolerance = more robust mechanism', 'Interface Uniformity U_Δ','Goldilocks Window Width (meV)') ax.set_xlim(0,1.0); ax.legend(fontsize=8.5,framealpha=0.9) ax.text(0.02,0.88,'Patterned nanoribbons\ntarget this regime →', transform=ax.transAxes,fontsize=8,color='#1a6b8a',fontweight='bold') ax = axes[1] U_arr2 = np.linspace(0.01,1.00,200) peak_dtc = [] for u in U_arr2: tc_a = np.array([tc_effective(d,u) for d in DEX_ARR]) peak_dtc.append(max(0, tc_a.max()-TC0)) peak_dtc = np.array(peak_dtc) ax.plot(U_arr2, peak_dtc, color=C['base'], lw=2.8, label='Peak ΔTc vs uniformity') ax.fill_between(U_arr2, 0, peak_dtc, alpha=0.15, color=C['base']) ax.axhline(0.5, color='gray', lw=1.2, ls=':', alpha=0.7, label='Paper lower bound (0.5K)') ax.axhline(2.0, color='gray', lw=1.2, ls=':', alpha=0.7, label='Paper upper bound (2.0K)') ax.fill_between(U_arr2, 0.5, 2.0, alpha=0.07, color='gray', label='Paper predicted range') for u, col, lbl in [(0.40,'#e67e22','U=0.40'),(0.70,C['good'],'U=0.70'),(0.95,'#1a6b8a','U=0.95')]: dtc = peak_dtc[np.argmin(np.abs(U_arr2-u))] ax.scatter([u],[dtc],color=col,s=80,zorder=6) ax.annotate(f'{lbl}: ΔTc=+{dtc:.2f}K',xy=(u,dtc),xytext=(u-0.15,dtc+0.1), fontsize=7.5,color=col,arrowprops=dict(arrowstyle='->',color=col)) sax(ax,'Figure 11b: Peak ΔTc vs Interface Uniformity\nMechanism enters paper\'s predicted range at U_Δ ≳ 0.55', 'U_Δ','Peak ΔTc (K)') ax.set_xlim(0,1.0); ax.set_ylim(-0.1,peak_dtc.max()*1.2+0.1) ax.legend(fontsize=8,framealpha=0.9) wm(fig11); plt.tight_layout(rect=[0,0.02,1,1]) fig11.savefig(os.path.join(OUT_DIR,'Fig11_Goldilocks_vs_Uniformity.png'),dpi=180,bbox_inches='tight',facecolor=C['bg']) plt.close(); print(" ✓ Fig11") # ── Fig 12: Tc robustness landscape ────────────────────────────────────────── fig12, axes = plt.subplots(1,2,figsize=(14,6)) fig12.patch.set_facecolor(C['bg']) # Left: 2D contour of Tc(Δ_mean, U_Δ) dex_g = np.linspace(0,DELTA_EV*4,150) u_g = np.linspace(0.05,1.0,150) DEX_G,U_G = np.meshgrid(dex_g,u_g) TC_G = np.zeros_like(DEX_G) for i in range(U_G.shape[0]): for j in range(DEX_G.shape[1]): TC_G[i,j] = tc_effective(DEX_G[i,j], U_G[i,j]) ax = axes[0] cf = ax.contourf(DEX_G*1000, U_G, TC_G, levels=20, cmap='RdYlGn') plt.colorbar(cf,ax=ax,label='Tc (K)') cs = ax.contour(DEX_G*1000,U_G,TC_G,levels=[TC0,TC0+0.5,TC0+1.0,TC0+1.5,TC0+2.0], colors='white',lw=1.0,alpha=0.8) ax.clabel(cs,fmt='%.1fK',fontsize=7.5,colors='white') ax.axvline(DELTA_EV*1000,color='white',lw=1.5,ls='--',alpha=0.8,label='~Pauli threshold') ax.axhline(0.40,color='#e67e22',lw=1.5,ls=':',alpha=0.9,label='Rough film (U≈0.4)') ax.axhline(0.95,color='#1a6b8a',lw=1.5,ls=':',alpha=0.9,label='Nanoribbon est. (U≈0.95)') sax(ax,'Figure 12a: Robustness Landscape — Tc(Δ_mean, U_Δ)\nGreen = above Tc₀; white contours show ΔTc levels', 'Mean Δ_ex (meV)','Interface Uniformity U_Δ') ax.legend(fontsize=7.5,loc='upper left',framealpha=0.8) # Right: Tc vs σ_Δ at fixed optimal mean ax = axes[1] sigma_mev = sigma_range*1000 ax.plot(sigma_mev, tc_vs_sigma, color=C['base'], lw=2.8, label=f'Tc at Δ_mean={dex_opt_perfect*1000:.2f}meV (optimal for U=1)') ax.axhline(TC0, color='black', lw=1.5, ls='--', alpha=0.7, label=f'Bare Tc₀={TC0}K') ax.fill_between(sigma_mev, TC0, tc_vs_sigma, where=tc_vs_sigma>TC0, alpha=0.15, color=C['base'], label='Region: scaffold beneficial') ax.fill_between(sigma_mev, tc_vs_sigma, TC0, where=tc_vs_sigma',color='#2e8b57',lw=1.5)) ax.text(0.50,0.35,'Uniform Δ_ex (each ribbon\nsame coupling strength)', ha='center',fontsize=7.5,color='#2e8b57',fontweight='bold') layers_R_top = [ (0.56,'#e8daef','Top hBN cap (~10nm)',0.06), (0.64,'#2c3e50','Graphite top gate',0.08), ] for y,col,lbl,h in layers_R_top: rect=Rectangle((0.05,y),0.90,h,lw=1,edgecolor='#333',facecolor=col,alpha=0.85) ax.add_patch(rect) ax.text(0.50,y+h/2,lbl,ha='center',va='center',fontsize=7.5, color='white' if col=='#2c3e50' else '#333') ax.text(0.50,0.82,'Proposed: Patterned Cobalt Nanoribbon Array',ha='center',va='center', fontsize=11,fontweight='bold',color='#1F3864') ax.text(0.50,0.77,'✓ Additional engineering parameters: N, width w,\nfill factor. Independent optimisation of U_Δ.', ha='center',va='center',fontsize=8.5,color='#2e8b57', bbox=dict(boxstyle='round',facecolor='#f0fff4',alpha=0.9)) axes[1].set_title('(b) Proposed Design — Patterned Cobalt Nanoribbon Array\nImproved Δ_ex spatial uniformity via geometry engineering', fontsize=10,fontweight='bold',color='#1F3864',pad=8) wm(fig13); plt.tight_layout(rect=[0,0.02,1,1]) fig13.savefig(os.path.join(OUT_DIR,'Fig13_Nanoribbon_Schematic.png'),dpi=180,bbox_inches='tight',facecolor=C['bg']) plt.close(); print(" ✓ Fig13") # ── Fig 14: f_pb and spacer sweep robustness ────────────────────────────────── fig14, axes = plt.subplots(1,2,figsize=(14,6)) fig14.patch.set_facecolor(C['bg']) ax = axes[0] u_arr = np.linspace(0.01,1.00,300) fpb_arr = np.array([frac_pair_broken(dex_opt_perfect,u) for u in u_arr]) ax.plot(u_arr, fpb_arr*100, color=C['warn'], lw=2.8, label='Fraction of k-space pair-broken') ax.fill_between(u_arr,0,fpb_arr*100,alpha=0.15,color=C['warn']) ax.axhline(10,color='gray',lw=1.2,ls=':',alpha=0.7,label='10% threshold (approx. tolerable)') ax.axvline(0.70,color=C['good'],lw=1.5,ls=':',label='U=0.70 (nanoribbon lower est.)') ax.axvline(0.95,color='#1a6b8a',lw=1.5,ls=':',label='U=0.95 (nanoribbon upper est.)') sax(ax,f'Figure 14a: Fraction of k-Space Exceeding Pair-Breaking Threshold vs U_Δ\nAt optimal mean Δ_ex={dex_opt_perfect*1000:.2f}meV', 'Interface Uniformity U_Δ','k-Space Pair-Broken Fraction (%)') ax.legend(fontsize=8,framealpha=0.9) ax = axes[1] u_vals_sp = [0.40, 0.70, 0.95, 1.00] sp_colors = ['#e67e22','#2e8b57','#1a6b8a','#2E5FA3'] sp_labels = ['U=0.40 (rough film)','U=0.70 (moderate)','U=0.95 (nanoribbon)','U=1.00 (perfect)'] for u,col,lbl in zip(u_vals_sp,sp_colors,sp_labels): tc_sp = np.array([tc_effective(dex_n(n),u) for n in SPACER_N]) ax.plot(SPACER_N,tc_sp,color=col,lw=2.2,marker='o',ms=7,label=lbl) ax.axhline(TC0,color='black',lw=1.5,ls='--',alpha=0.7,label=f'Bare Tc₀={TC0}K') sax(ax,'Figure 14b: Tc vs hBN Spacer Thickness at Multiple Uniformity Levels\nOptimal spacer may shift with interface quality', 'hBN Layers (n)','Tc (K)') ax.set_xticks(SPACER_N) ax.set_xticklabels([f'n={n}' for n in SPACER_N]) ax.legend(fontsize=8,framealpha=0.9) wm(fig14); plt.tight_layout(rect=[0,0.02,1,1]) fig14.savefig(os.path.join(OUT_DIR,'Fig14_Robustness_Spacer.png'),dpi=180,bbox_inches='tight',facecolor=C['bg']) plt.close(); print(" ✓ Fig14") # ───────────────────────────────────────────────────────────────────────────── # SECTION 5 — SUMMARY # ───────────────────────────────────────────────────────────────────────────── # Find U threshold where ΔTc enters paper range u_threshold_low = U_arr2[np.argmin(np.abs(peak_dtc-0.5))] if (peak_dtc>=0.5).any() else 1.0 u_threshold_high = U_arr2[np.argmin(np.abs(peak_dtc-2.0))] if (peak_dtc>=2.0).any() else 1.0 win_at_95 = window_widths[np.argmin(np.abs(U_sweep-0.95))] win_at_40 = window_widths[np.argmin(np.abs(U_sweep-0.40))] summary = f""" ================================================================================ SIMULATION SUMMARY v3.0 — Topological Cooper Scaffold Scientific question: Robustness to interface non-uniformity Run: {datetime.now().strftime('%Y-%m-%d %H:%M:%S')} ================================================================================ CALIBRATION ──────────────────────────────────────────────────────────────────────────────── λ₀ (Allen-Dynes): {LAM0:.4f} Tc₀ check: {ad_base(LAM0):.4f}K ✓ ~Pauli threshold: {DELTA_EV*1000:.3f} meV Optimal mean Δ_ex (U=1.0): {dex_opt_perfect*1000:.3f} meV UNIFORMITY PARAMETER U_Δ ──────────────────────────────────────────────────────────────────────────────── U_Δ = 1.0: perfectly uniform (ideal limit) U_Δ = 0.7-0.9: patterned nanoribbon array (estimated) U_Δ = 0.3-0.5: continuous cobalt film (estimated rough) U_Δ = 0.0: completely random Δ_ex (no spatial control) Physical model: σ_Δ = (1 - U_Δ) × Δ_mean Pair-broken fraction: f_pb = P(Δ_ex,local > Δ_pb) [Gaussian tail] Effective Tc: Tc_eff = Tc_uniform × (1 - α × f_pb), α = {ALPHA_PB} PEAK ΔTc AT EACH UNIFORMITY LEVEL ──────────────────────────────────────────────────────────────────────────────── U_Δ Peak ΔTc (K) Goldilocks width (meV) In predicted range [0.5,2.0K]? {''.join([f"{u:.2f} +{results[u]['dtc']:.3f}K {results[u]['win_width_mev']:.3f} meV {'YES ✓' if 0.5<=results[u]['dtc']<=2.0 else 'NO'}{chr(10)}" for u in U_LEVELS])} ROBUSTNESS FINDINGS ──────────────────────────────────────────────────────────────────────────────── U_Δ threshold for ΔTc ≥ 0.5K (paper lower bound): ~{u_threshold_low:.2f} Goldilocks window width at U=0.40 (rough film): {win_at_40:.3f} meV Goldilocks window width at U=0.95 (nanoribbons): {win_at_95:.3f} meV Window width improvement (0.40→0.95): {win_at_95/max(win_at_40,1e-6):.1f}× wider NANORIBBON GEOMETRY ADVANTAGES (independent of Litz/Roebel analogy) ──────────────────────────────────────────────────────────────────────────────── 1. Additional fabrication parameters: N (count), w (width), fill factor 2. Tunable spatial frequency of Δ_ex modulation (at moiré unit cell scale) 3. Reduced strain from continuous cobalt film 4. Better interface quality per nanoribbon vs continuous film 5. Independent optimisation of U_Δ separate from mean Δ_ex 6. Nanoribbon count N sweep is new testable prediction (PR2) KEY SCIENTIFIC RESULT (RECOMMENDED PAPER WORDING) ──────────────────────────────────────────────────────────────────────────────── "The robustness of the Topological Cooper Scaffold mechanism to realistic interface non-uniformity was investigated by introducing an interface uniformity parameter U_Δ ∈ [0,1], where U_Δ = 1 represents a perfectly uniform exchange field and smaller values represent increasing spatial variance. The fraction of k-space where the local exchange splitting exceeds the pair-breaking threshold increases rapidly as U_Δ decreases. The simulation shows that the scaffold mechanism produces ΔTc within the predicted [0.5, 2.0] K range for U_Δ ≳ {u_threshold_low:.2f}. A continuous cobalt film is estimated to achieve U_Δ ≈ 0.3–0.5, which may produce only modest ΔTc. Patterned cobalt nanoribbon arrays, which allow independent control of spatial Δ_ex distribution through geometric parameters (ribbon count N, width w, fill factor), are estimated to achieve U_Δ ≈ 0.7–0.9, potentially widening the effective Goldilocks window by {win_at_95/max(win_at_40,1e-6):.1f}× and bringing ΔTc into the paper's predicted range. These results motivate patterned nanoribbon geometry as a future-work direction, independent of any cross-domain analogy." DISCLAIMER ──────────────────────────────────────────────────────────────────────────────── U_Δ is a new parameter requiring experimental calibration. Estimated U_Δ ranges for film vs nanoribbon geometries are order-of-magnitude estimates only. The simulation framework is semi-microscopic, not DFT. α_pb (Tc sensitivity to pair-broken k-space fraction) = {ALPHA_PB} is a model assumption requiring validation. ================================================================================ """ with open(os.path.join(OUT_DIR,'cooper_scaffold_summary_v3.txt'),'w') as f: f.write(summary) print(summary) print(f"\nAll outputs saved to: {OUT_DIR}")